Directional Ballistic Transport for Partially Periodic Schrödinger Operators on $\mathbb{Z}^2$

TL;DR

This work analyzes Schrödinger operators $H=H_0+V$ on $\mathbb{Z}^2$ with strip-periodic potentials, focusing on surface states that live near the strip and scattering states that radiate away. By developing a partial Floquet framework and a transfer-matrix approach, the authors establish analytic variation of embedded surface-band energies $\lambda_{l,i}(k)$ across the Brillouin zone and prove that surface states exhibit directional ballistic transport: a nonzero asymptotic velocity along the periodic direction and zero along the transverse direction, with a purely absolutely continuous spectrum on the surface subspace. They also show that a dense subset of scattering states are ballistic and prove density of the natural domain $D(\vec{Q})$ in the surface subspace, ensuring these transport properties hold on a large portion of the state space. Collectively, the results provide a quantitative, anisotropic transport description for partially periodic, higher-dimensional Schrödinger operators and extend ballistic transport phenomena beyond fully periodic or one-dimensional settings.

Abstract

We study the transport properties of Schrödinger operators on $\mathbb{Z}^2$ with potentials that are periodic in one direction and compactly supported in the other. Such systems are known to produce surface states that are weakly confined near the support of the potential. We demonstrate that surface states exhibit what we describe as directional ballistic transport, characterized by a strong form of ballistic transport along the periodic direction and its absence in the compactly supported one. By showing that the scattering states exhibit ballistic transport, we obtain ballistic transport for a dense subset of all of $\ell^2(\mathbb {Z} ^2)$.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 3.1
• proof
• Proposition 3.2